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Question Number 140534 Answers: 1 Comments: 0

Question Number 140533 Answers: 1 Comments: 0

A triangle is inscribed in a circle. the vertices of the triangle divided the circumference of the circle into three area of length 6,8,10 units then the area of triangle is equal to... (a) ((64(√3)((√3)+1))/π^2 ) (c) ((36(√3)((√3)−1))/π^2 ) (b) ((72(√3)((√3)+1))/π^2 ) (d) ((36(√3)((√3)+1))/π^2 )

$$\mathrm{A}\:\mathrm{triangle}\:\mathrm{is}\:\mathrm{inscribed}\:\mathrm{in}\:\mathrm{a}\:\mathrm{circle}. \\ $$$$\mathrm{the}\:\mathrm{vertices}\:\mathrm{of}\:\mathrm{the}\:\mathrm{triangle}\:\mathrm{divided} \\ $$$$\mathrm{the}\:\mathrm{circumference}\:\mathrm{of}\:\mathrm{the}\:\mathrm{circle} \\ $$$$\mathrm{into}\:\mathrm{three}\:\mathrm{area}\:\mathrm{of}\:\mathrm{length}\:\mathrm{6},\mathrm{8},\mathrm{10} \\ $$$$\mathrm{units}\:\mathrm{then}\:\mathrm{the}\:\mathrm{area}\:\mathrm{of}\:\mathrm{triangle} \\ $$$$\mathrm{is}\:\mathrm{equal}\:\mathrm{to}... \\ $$$$\left(\mathrm{a}\right)\:\frac{\mathrm{64}\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{3}}+\mathrm{1}\right)}{\pi^{\mathrm{2}} }\:\:\:\:\left(\mathrm{c}\right)\:\frac{\mathrm{36}\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{3}}−\mathrm{1}\right)}{\pi^{\mathrm{2}} } \\ $$$$\left(\mathrm{b}\right)\:\frac{\mathrm{72}\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{3}}+\mathrm{1}\right)}{\pi^{\mathrm{2}} }\:\left(\mathrm{d}\right)\:\frac{\mathrm{36}\sqrt{\mathrm{3}}\left(\sqrt{\mathrm{3}}+\mathrm{1}\right)}{\pi^{\mathrm{2}} } \\ $$

Question Number 140531 Answers: 1 Comments: 0

Prove that ∫^( x) _0 (t/(e^t −1)) dt = Σ_(n=1) ^(+∞) (((1−e^(−x) )^n )/n^2 )

$$\mathrm{Prove}\:\mathrm{that}\:\:\underset{\mathrm{0}} {\int}^{\:\mathrm{x}} \:\frac{\mathrm{t}}{\mathrm{e}^{\mathrm{t}} −\mathrm{1}}\:\mathrm{dt}\:=\:\underset{\mathrm{n}=\mathrm{1}} {\overset{+\infty} {\sum}}\:\frac{\left(\mathrm{1}−\mathrm{e}^{−\mathrm{x}} \right)^{\mathrm{n}} }{\mathrm{n}^{\mathrm{2}} } \\ $$

Question Number 140530 Answers: 1 Comments: 0

lim_(x→0) (((x+y)sec (x+y)−ysec y)/x)=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\left(\mathrm{x}+\mathrm{y}\right)\mathrm{sec}\:\left(\mathrm{x}+\mathrm{y}\right)−\mathrm{ysec}\:\mathrm{y}}{\mathrm{x}}=? \\ $$

Question Number 140529 Answers: 1 Comments: 0

Question Number 140525 Answers: 2 Comments: 0

x;y;z>0 ; x+y+z=3 proof (x^3 +2)(y^3 +2)(z^3 +2)≥3^3

$${x};{y};{z}>\mathrm{0}\:;\:{x}+{y}+{z}=\mathrm{3} \\ $$$${proof}\:\left({x}^{\mathrm{3}} +\mathrm{2}\right)\left({y}^{\mathrm{3}} +\mathrm{2}\right)\left({z}^{\mathrm{3}} +\mathrm{2}\right)\geqslant\mathrm{3}^{\mathrm{3}} \\ $$

Question Number 140652 Answers: 1 Comments: 0

Question Number 140519 Answers: 0 Comments: 0

Question Number 140513 Answers: 0 Comments: 0

n ∈ N^∗ and k ∈ N^∗ . Given 0≤k≤n−1. Show that (1/n)ln(1+(k/n))≤∫_(1+(k/n)) ^(1+((k+1)/n)) lnx dn≤(1/n)ln(1+((k+1)/n))

$$\mathrm{n}\:\in\:\mathbb{N}^{\ast} \:\mathrm{and}\:\mathrm{k}\:\in\:\mathbb{N}^{\ast} . \\ $$$$\mathrm{Given}\:\mathrm{0}\leqslant\mathrm{k}\leqslant\mathrm{n}−\mathrm{1}. \\ $$$$\mathrm{Show}\:\mathrm{that}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{n}}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}\right)\leqslant\underset{\mathrm{1}+\frac{\mathrm{k}}{\mathrm{n}}} {\overset{\mathrm{1}+\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}} {\int}}\mathrm{lnx}\:\mathrm{dn}\leqslant\frac{\mathrm{1}}{\mathrm{n}}\mathrm{ln}\left(\mathrm{1}+\frac{\mathrm{k}+\mathrm{1}}{\mathrm{n}}\right) \\ $$

Question Number 140506 Answers: 2 Comments: 0

e^((((ζ(2))/2)−((ζ(3))/3)+((ζ(4))/4)−((ζ(5))/5)+...)) =?

$${e}^{\left(\frac{\zeta\left(\mathrm{2}\right)}{\mathrm{2}}−\frac{\zeta\left(\mathrm{3}\right)}{\mathrm{3}}+\frac{\zeta\left(\mathrm{4}\right)}{\mathrm{4}}−\frac{\zeta\left(\mathrm{5}\right)}{\mathrm{5}}+...\right)} \:\:\:=? \\ $$

Question Number 140503 Answers: 0 Comments: 3

Prove it (√(a+(√b)))=(√(((a+(√(a^2 +b)))/2)+))(√((a−(√(a^2 −b)))/2))

$$\boldsymbol{\mathrm{P}}\mathrm{rove}\:\mathrm{it} \\ $$$$\sqrt{\boldsymbol{{a}}+\sqrt{\boldsymbol{{b}}}}=\sqrt{\frac{\boldsymbol{{a}}+\sqrt{\boldsymbol{{a}}^{\mathrm{2}} +\boldsymbol{{b}}}}{\mathrm{2}}+}\sqrt{\frac{\boldsymbol{{a}}−\sqrt{\boldsymbol{{a}}^{\mathrm{2}} −\boldsymbol{{b}}}}{\mathrm{2}}} \\ $$

Question Number 140502 Answers: 0 Comments: 0

If three vector a^→ , b^→ and c^→ are such that a^→ ≠ 0 and a^→ ×b^→ = 2(a^→ ×c^→ ) ,∣a^→ ∣ = ∣c^→ ∣ = 1 , ∣b^→ ∣ = 4 and the angle between b^→ and c^→ is cos^(−1) ((1/4)), then b^→ −2c^→ = λ a^→ , where λ =?

Question Number 140500 Answers: 0 Comments: 0

tan^2 1°+tan^2 2°+tan^2 3°+...+tan^2 89°=((15931)/3) ???

$$\mathrm{tan}\:^{\mathrm{2}} \mathrm{1}°+\mathrm{tan}\:^{\mathrm{2}} \mathrm{2}°+\mathrm{tan}\:^{\mathrm{2}} \mathrm{3}°+...+\mathrm{tan}\:^{\mathrm{2}} \mathrm{89}°=\frac{\mathrm{15931}}{\mathrm{3}}\:\:\:\:\:??? \\ $$

Question Number 140498 Answers: 2 Comments: 0

........ nice ....... calculus....... simplify: 𝛗(x):= sin((x/2))(1+2Σ_(m=1) ^n cos(mx))

$$\:\:\:\:\:\:........\:{nice}\:\:\:.......\:\:{calculus}....... \\ $$$$\:\:\:\:\:\:{simplify}: \\ $$$$\:\:\:\:\boldsymbol{\phi}\left({x}\right):=\:{sin}\left(\frac{{x}}{\mathrm{2}}\right)\left(\mathrm{1}+\mathrm{2}\underset{{m}=\mathrm{1}} {\overset{{n}} {\sum}}{cos}\left({mx}\right)\right) \\ $$$$\:\:\:\: \\ $$

Question Number 140497 Answers: 1 Comments: 7

sin^(−1) (sin x)=x sin^(−1) (cos x)=? sin^(−1) (tan x)=?

$$\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{sin}\:\mathrm{x}\right)=\mathrm{x} \\ $$$$\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{cos}\:\mathrm{x}\right)=? \\ $$$$\mathrm{sin}^{−\mathrm{1}} \left(\mathrm{tan}\:\mathrm{x}\right)=? \\ $$

Question Number 140471 Answers: 2 Comments: 0

If lim_(x→0) (cos x + a sin bx)^(1/x) = e^2 { ((a=?)),((b=?)) :}

$$\mathrm{If}\:\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\left(\mathrm{cos}\:\mathrm{x}\:+\:\mathrm{a}\:\mathrm{sin}\:\mathrm{bx}\right)^{\frac{\mathrm{1}}{\mathrm{x}}} \:=\:\mathrm{e}^{\mathrm{2}} \\ $$$$\:\begin{cases}{\mathrm{a}=?}\\{\mathrm{b}=?}\end{cases} \\ $$